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Article Excerpt
Mean-variance portfolios constructed using the sample mean and covariance matrix of asset returns perform poorly out of sample due to estimation error. Moreover, it is commonly accepted that estimation error in the sample mean is much larger than in the sample covariance matrix. For this reason, researchers have recently focused on the minimum-variance portfolio, which relies solely on estimates of the covariance matrix, and thus usually performs better out of sample. However, even the minimum-variance portfolios are quite sensitive to estimation error and have unstable weights that fluctuate substantially over time. In this paper, we propose a class of portfolios that have better stability properties than the traditional minimum-variance portfolios. The proposed portfolios are constructed using certain robust estimators and can be computed by solving a single nonlinear program, where robust estimation and portfolio optimization arc performed in a single step. We show analytically that the resulting portfolio weights are less sensitive to changes in the asset-return distribution than those of the traditional portfolios. Moreover, our numerical results on simulated and empirical data confirm that the proposed portfolios are more stable than the traditional minimum-variance portfolios, while preserving (or slightly improving) their relatively good out-of-sample performance.
Subject classifications: finance: portfolio, investment; economics: econometrics; portfolio choice; minimum-variance portfolios; estimation error; robust statistics.
Area of review: Financial Engineering.
History: Received February 2007; revisions received August 2007, November 2007, December 2007; accepted December 2007. Published online in Articles in Advance February 26, 2009.
1. Introduction
An investor who cares only about the mean and variance of static portfolio returns should hold a portfolio on the mean-variance efficient frontier, which was first characterized by Markowitz (1952). To implement these portfolios in practice, one has to estimate the mean and the covariance matrix of asset returns. Traditionally, the sample mean and covariance matrix have been used for this purpose. However, because of estimation error, policies constructed using these estimators are extremely unstable; that is, the resulting portfolio weights fluctuate substantially over time. This has greatly undermined the popularity of mean-variance portfolios among portfolio managers, who are reluctant to implement policies that recommend such drastic changes in the portfolio composition. Moreover, the concerns of portfolio managers are reinforced by well-known empirical evidence, which shows that these unstable portfolios perform very poorly in terms of their out-of-sample mean and variance; see Michaud (1989), Chopra and Ziemba (1993), and Broadie (1993).
The instability of the mean-variance portfolios can be explained (partly) by the well-documented difficulties associated with estimating mean asset returns; see Merton (1980). For this reason, researchers have recently focused on the minimum-variance portfolio, which relies solely on estimates of the covariance matrix, and thus is not as sensitive to estimation error (Chan et al. 1999, Jagannathan and Ma 2003). Jagannathan and Ma, for example, state that "the estimation error in the sample mean is so large that nothing much is lost in ignoring the mean altogether" (p. 1652). This claim is substantiated by extensive empirical evidence that shows the minimum-variance portfolio usually performs better out of sample than any other mean-variance portfolio--even when Sharpe ratio or other performance measures related to both the mean and variance are used for the comparison; see Jorion (1986), Jagannathan and Ma (2003), and DeMiguel et al. (2005). Moreover, in this paper we provide numerical results that also illustrate the perils associated with using estimates of mean returns for portfolio selection. For all these reasons, herein our discussion focuses on the minimum-variance portfolios.
Although the minimum-variance portfolio does not rely on estimates of mean returns, it is still quite vulnerable to the impact of estimation error; see Chan et al. (1999) and Jagannathan and Ma (2003). The sensitivity of the minimum-variance portfolio to estimation error is surprising. These portfolios are based on the sample covariance matrix, which is the maximum likelihood estimator (MLE) for normally distributed returns. Moreover, MLEs are theoretically the most efficient for the assumed distribution; that is, these estimators have the smallest asymptotic variance provided the data follows the assumed distribution. So why does the sample covariance matrix give unstable portfolios? The answer is the efficiency of MLEs based on assuming normality of returns is highly sensitive to deviations of the asset-return distribution from the assumed (normal) distribution. In particular, MLEs based on the normality assumption are not necessarily the most efficient for data that depart even slightly from normality; see Example 1.1 in Huber (2004). This is particularly important for portfolio selection, where extensive evidence shows that the empirical distribution of returns usually deviates from the normal distribution.
To induce greater stability on the minimum-variance portfolio weights, in this paper we propose a class of policies that arc constructed using robust estimators of the portfolio return characteristics. A robust estimator is one that gives meaningful information about asset returns even when the empirical (sample) distribution deviates from the assumed (normal) distribution (see Huber 2004, Hampel et al. 1986, Rousseeuw and Leroy 1987). Specifically, a robust estimator should have good properties not only for the assumed distribution, but also for any distribution in a neighborhood of the assumed one.
Classical examples of robust estimators are the median and the mean absolute deviation (MAD). The median is the value that is larger than 50% and smaller than 50% of the sample data points whereas the MAD is the mean absolute deviation from the median. The following example from Tukey (1960) illustrates the advantages of using robust estimators. Assume that all but a small fraction h of the data are drawn from a univariate normal distribution, whereas the remainder are drawn from the same normal distribution, but with a standard deviation three times larger. Then, a value of h = 10% is enough to make the median as efficient as the mean, whereas more sophisticated robust estimators are 40% more efficient than the mean with h = 10%. Moreover, even h = 0.1% is enough to make the MAD more efficient than the standard deviation. The conclusion is that when the sample distribution deviates even slightly from the assumed distribution, the efficiency of classical estimators may be drastically reduced. Robust estimators, on the other hand, are not as efficient as MLEs when the underlying model is correct, but their properties are not as sensitive to deviations from the assumed distribution.
For this reason, we examine portfolio policies based on robust estimators. These policies should be less sensitive to deviations of the empirical distribution of returns from normality than the traditional policies. We focus on certain robust estimators known as M- and S-estimators, which have better properties than the classical median and MAD.
Our paper makes three contributions. Our first contribution is to show how one can compute the portfolio policy that minimizes a robust estimator of risk by solving a single nonlinear program. As mentioned above, we focus on minimum-risk portfolios because they usually perform better out of sample than portfolios that optimize the trade-off between in-sample risk and return. The proposed portfolios are the solution to a nonlinear program where portfolio optimization and robust estimation are performed in a single step. In particular, the decision variables of this optimization problem are the portfolio weights, and its objective is either the M- or S-estimator of portfolio risk.
Our second contribution is to characterize (analytically) the properties of the resulting portfolios. Specifically, we give an analytical bound on the sensitivity of the portfolio weights to changes in the distribution of asset returns. Our analysis shows that the portfolio weights of the proposed policies are less sensitive to changes in the distributional assumptions than those of the traditional minimum-variance policies. As a result, the portfolio weights of the proposed policies are more stable than those of the traditional policies. This makes the proposed portfolios a credible alternative to the traditional policies in the eyes of the investors, who are usually reticent to implement portfolios whose recommended weights fluctuate substantially over time.
Our third contribution is to compare the behavior of the proposed portfolios to that of the traditional portfolios on simulated and empirical data. The results confirm that minimum-risk portfolios (standard and robust) attain higher out-of-sample Sharpe ratios than return-risk portfolios (standard and robust). As mentioned above, this is because estimates of mean returns (standard and robust) contain so much estimation error that using them for portfolio selection worsens performance. Comparing the proposed minimum-risk portfolios to the traditional minimum-variance portfolios, we observe that the proposed portfolios have more stable weights than the traditional portfolios, while preserving (or slightly improving) their high out-of-sample Sharpe ratios.
Other researchers have proposed portfolio policies based on robust estimation techniques; see Cavadini et al. (2001), Vaz-de Melo and Camara (2003), Perret-Gentil and Victoria-Feser (2004), and Welsch and Zhou (2007). Their approaches, however, differ from ours. All three papers compute the robust portfolio policies in two steps. First, they compute a robust estimate of the covariance matrix of asset returns. Second, they solve the minimum-variance problem where the covariance matrix is replaced by its robust estimate. We, on the other hand, propose solving a single nonlinear program, where portfolio optimization and robust estimation are performed in one step.
The only other one-step approach to robust portfolio estimation is in Lauprete et al. (2002); see also Lauprete (2001). They consider a one-step robust approach based on the M-estimator of risk and give some numerical results. We, in addition, consider portfolios based on the S-estimators, give an analytical bound on the sensitivity of the M- and S-portfolio weights to changes in the distributional assumptions, and examine the behavior of both the M- and S-portfolios on simulated and empirical data sets.
Finally, a number of other approaches have been proposed in the literature to address estimation error. The robust portfolio optimization approach (see, for example, Goldfarb and Iyengar 2003, Tutuncu and Koenig 2004, Garlappi et al. 2007, Lu 2006) explicitly recognizes that the result of the estimation process is not a single-point estimate, but rather an uncertainty set, where the true mean and covariance matrix of asset returns lie with certain confidence. A robust portfolio is, then, one that optimizes the worst-case performance with respect to all possible values the mean and covariance matrix may take within their corresponding uncertainty sets. Bayesian portfolio policies are constructed using estimators that are generated by combining the investor's prior beliefs with the evidence obtained from historical return data; see Jorion (1986), Black and Litterman (1992), and Pastor and Stambaugh (2000). Finally, Jagannathan and Ma (2003) show that imposing short-selling constraints can help to reduce the impact of estimation error on the stability and performance of the minimum-variance portfolio.
The rest of this paper is organized as follows. Section 2 reviews the mean-variance and minimum-variance portfolios and highlights their lack of stability with a simple example. In [section]3, we show how to compute the M- and S-portfolios. In [section]4, we analyze the sensitivity of the proposed portfolio policies to changes in the empirical distribution of asset returns. In [section]5, we compare the different policies on simulated and empirical data. Section 6 concludes.
2. On the Instability of the Traditional Portfolios
In this section, we use a simple example to illustrate the instability of the portfolio weights of the mean-variance and minimum-variance policies. In particular, we consider two risky assets whose returns follow a normal distribution most of the time, but there is a small probability that the returns of the two risky assets follow a different deviation distribution. That is, we assume that the true asset-return distribution is
G = 99% X N([mu],[SIGMA]) + 1% X D, (1)
where N([mu], [SIGMA]) is a normal distribution with mean [mu] and covariance matrix [SIGMA], and D is a deviation distribution. Specifically, we are going to consider the case where there is a 99% probability that the returns of the two assets are independently and identically distributed following a normal distribution with an annual mean of 12% and an annual standard deviation of 16%, and there is a 1% probability that the returns of the two assets are distributed according to a normal distribution with the same covariance matrix but with the mean return for the second asset equal to --50 times the mean return of the first asset. That is, we assume that h = 1%,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and D = N([[mu].sub.d], [[SIGMA].sub.d])), where [[SIGMA].sub.d] = [SIGMA] and [[mu].sub.d] = (0.01, -0.50).
Finally, we would like to note that a basic assumption of our work is that the investor does...
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